| L(s) = 1 | − 4·2-s + 6·3-s + 4·4-s − 4·5-s − 24·6-s + 16·8-s + 13·9-s + 16·10-s + 24·12-s − 12·13-s − 24·15-s − 64·16-s − 32·17-s − 52·18-s − 42·19-s − 16·20-s − 12·23-s + 96·24-s + 24·25-s + 48·26-s + 6·27-s − 4·29-s + 96·30-s + 64·32-s + 128·34-s + 52·36-s − 96·37-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 4/5·5-s − 4·6-s + 2·8-s + 13/9·9-s + 8/5·10-s + 2·12-s − 0.923·13-s − 8/5·15-s − 4·16-s − 1.88·17-s − 2.88·18-s − 2.21·19-s − 4/5·20-s − 0.521·23-s + 4·24-s + 0.959·25-s + 1.84·26-s + 2/9·27-s − 0.137·29-s + 16/5·30-s + 2·32-s + 3.76·34-s + 13/9·36-s − 2.59·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4042951800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4042951800\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 42 T + 47 p T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T + 23 T^{2} - 22 p T^{3} + 148 T^{4} - 22 p^{3} T^{5} + 23 p^{4} T^{6} - 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} - 104 T^{3} - 449 T^{4} - 104 p^{2} T^{5} - 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 4482 T^{4} - 80 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 p T^{2} + 26403 T^{4} + 2 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T - 133 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 32 T + 310 T^{2} + 256 p T^{3} + 451 p^{2} T^{4} + 256 p^{3} T^{5} + 310 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 308 T^{2} + 3120 T^{3} - 186849 T^{4} + 3120 p^{2} T^{5} + 308 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - 1640 T^{2} - 104 T^{3} + 2021599 T^{4} - 104 p^{2} T^{5} - 1640 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2264 T^{2} + 2540466 T^{4} - 2264 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 48 T + 2564 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 26 T - 2375 T^{2} + 8086 T^{3} + 5346484 T^{4} + 8086 p^{2} T^{5} - 2375 p^{4} T^{6} - 26 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 72 T + 4858 T^{2} - 225360 T^{3} + 9573171 T^{4} - 225360 p^{2} T^{5} + 4858 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3928 T^{2} + 10549503 T^{4} + 3928 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 124 T + 6394 T^{2} - 417136 T^{3} + 29594659 T^{4} - 417136 p^{2} T^{5} + 6394 p^{4} T^{6} - 124 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 78 T + 9487 T^{2} - 581802 T^{3} + 50578788 T^{4} - 581802 p^{2} T^{5} + 9487 p^{4} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 44 T - 5960 T^{2} + 19976 T^{3} + 41297119 T^{4} + 19976 p^{2} T^{5} - 5960 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 102 T + 8903 T^{2} - 554370 T^{3} + 24955956 T^{4} - 554370 p^{2} T^{5} + 8903 p^{4} T^{6} - 102 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 204 T + 25462 T^{2} + 2364360 T^{3} + 178845171 T^{4} + 2364360 p^{2} T^{5} + 25462 p^{4} T^{6} + 204 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 26 T - 9671 T^{2} - 8086 T^{3} + 75059764 T^{4} - 8086 p^{2} T^{5} - 9671 p^{4} T^{6} + 26 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 360 T + 65842 T^{2} - 8151120 T^{3} + 743321283 T^{4} - 8151120 p^{2} T^{5} + 65842 p^{4} T^{6} - 360 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 18170 T^{2} + 161815347 T^{4} - 18170 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 16 T - 1130 T^{2} - 231296 T^{3} - 62849021 T^{4} - 231296 p^{2} T^{5} - 1130 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 234 T + 22369 T^{2} + 3175146 T^{3} + 445189284 T^{4} + 3175146 p^{2} T^{5} + 22369 p^{4} T^{6} + 234 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.40151041902977604560963584289, −10.34251457766205531769950887157, −9.690817352857108886998415639908, −9.360156890356644938415447114751, −9.046077504966853677014751480462, −8.970064967434868018214504110193, −8.859917865425950050687457118237, −8.387625861817615471193328203327, −8.228353295556863592233915869468, −8.142813671897009567746790729330, −7.61904357394588394609833972769, −7.28784365161502871860754827825, −7.06988588267110842598170234306, −6.71025352613721274954752075195, −6.44263903231675646344323920329, −5.39290896132221353326462828956, −5.33097825056934813467706984828, −4.47200404833400817675958655885, −4.20015857325202758538814013116, −4.12200762065085217455916416857, −3.55392360214029619597113251437, −2.54045157984970087036309600163, −2.36644235769458789151022849659, −1.89725505461771836363961699614, −0.45502623186036427545334138825,
0.45502623186036427545334138825, 1.89725505461771836363961699614, 2.36644235769458789151022849659, 2.54045157984970087036309600163, 3.55392360214029619597113251437, 4.12200762065085217455916416857, 4.20015857325202758538814013116, 4.47200404833400817675958655885, 5.33097825056934813467706984828, 5.39290896132221353326462828956, 6.44263903231675646344323920329, 6.71025352613721274954752075195, 7.06988588267110842598170234306, 7.28784365161502871860754827825, 7.61904357394588394609833972769, 8.142813671897009567746790729330, 8.228353295556863592233915869468, 8.387625861817615471193328203327, 8.859917865425950050687457118237, 8.970064967434868018214504110193, 9.046077504966853677014751480462, 9.360156890356644938415447114751, 9.690817352857108886998415639908, 10.34251457766205531769950887157, 10.40151041902977604560963584289
